Question

1. below are graphs of $f(x) = |x|$ (thin line) and its translation (bold line). write an equation of the translation.

simplify the fraction by rationalizing the denominator.

1. $\frac{3}{sqrt{2}}$
2. $\frac{15}{4sqrt{5}}$

solve by factoring.

1. $2x^3 - 4x^2 - 126x = 0$
2. $30x^2 - 17x - 2 = 0$

Explanation:

Step1: Identify vertex of translated graph

The vertex of $f(x)=|x|$ is $(0,0)$. The bold translated graph has vertex $(0,4)$ and opens downward, so it is a reflection and vertical shift.

Step2: Write equation for translation

Reflect $|x|$ over x-axis: $-|x|$, then shift up 4 units: $g(x) = -|x| + 4$
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Step1: Rationalize $\frac{3}{\sqrt{2}}$

Multiply numerator/denominator by $\sqrt{2}$
$\frac{3 \times \sqrt{2}}{\sqrt{2} \times \sqrt{2}} = \frac{3\sqrt{2}}{2}$
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Step1: Rationalize $\frac{15}{4\sqrt{5}}$

Multiply numerator/denominator by $\sqrt{5}$
$\frac{15 \times \sqrt{5}}{4\sqrt{5} \times \sqrt{5}} = \frac{15\sqrt{5}}{4 \times 5} = \frac{3\sqrt{5}}{4}$
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Step1: Factor out GCF from $2x^3-4x^2-126x=0$

Factor $2x$: $2x(x^2 - 2x - 63) = 0$

Step2: Factor quadratic

Factor $x^2-2x-63$: $2x(x-9)(x+7)=0$

Step3: Solve for $x$

Set each factor to 0: $2x=0$, $x-9=0$, $x+7=0$
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Step1: Factor quadratic $30x^2-17x-2=0$

Find two terms: $30x^2+3x-20x-2=0$, group: $3x(10x+1)-2(10x+1)=0$ → $(3x-2)(10x+1)=0$

Step2: Solve for $x$

Set each factor to 0: $3x-2=0$, $10x+1=0$

Answer:

1. $g(x) = -|x| + 4$
2. $\frac{3\sqrt{2}}{2}$
3. $\frac{3\sqrt{5}}{4}$
4. $x=0$, $x=9$, $x=-7$
5. $x=\frac{2}{3}$, $x=-\frac{1}{10}$